## Abstract

There has been a great deal of recent interest in learning and approximation of functions that can be expressed as expectations of a given nonlinearity with respect to its random internal parameters. Examples of such representations include “infinitely wide” neural nets, where the underlying nonlinearity is given by the activation function of an individual neuron. In this paper, we bring this perspective to function representation by neural stochastic differential equations (SDEs). A neural SDE is an Itô diffusion process whose drift and diffusion matrix are elements of some parametric families. We show that the ability of a neural SDE to realize nonlinear functions of its initial condition can be related to the problem of optimally steering a certain deterministic dynamical system between two given points in finite time. This auxiliary system is obtained by formally replacing the Brownian motion in the SDE by a deterministic control input. We derive upper and lower bounds on the minimum control effort needed to accomplish this steering; these bounds may be of independent interest in the context of motion planning and deterministic optimal control.

Original language | English (US) |
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Pages (from-to) | 838-850 |

Number of pages | 13 |

Journal | Proceedings of Machine Learning Research |

Volume | 211 |

State | Published - 2023 |

Event | 5th Annual Conference on Learning for Dynamics and Control, L4DC 2023 - Philadelphia, United States Duration: Jun 15 2023 → Jun 16 2023 |

## Keywords

- function representation
- machine learning from a continuous viewpoint
- neural SDEs
- nonlinear controllability

## ASJC Scopus subject areas

- Artificial Intelligence
- Software
- Control and Systems Engineering
- Statistics and Probability